Though the mirror map is not defined for \(q=1\) , as the resulting \(\Phi \)  is not strictly convex, the limiting algorithm as \(q\rightarrow 1\)  corresponds to the so-called neg-entropy map in terms of regret: we elaborate on this in great detail in Section REF . OMD with a neg-entropy map amounts to a multiplicative update rule (see Eq. (REF )). Intermediate values of \(q\in (1,2]\)  thus establish a continuum [1]} between a multiplicative-update algorithm (OMD with a neg-entropy map) and an additive-update algorithm (OGD).
